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Heat transfer in composite media subject to distributed sources, and time-dependent discrete sources and surroundings
Heat
Transfer
Distributed
in Composite Media Subject to
Sources, and Time-Dependent
Discrete
Sources and Surroundings br
M. H.
COBBLE
Department of Mechanical Engineering New Mexico State Universip, Las Cruces, New Mexico The equation for the temperature
ABSTRACT:
is solved.
The composite
material
solidly
joined
distribution in each of k sections of a composite
consists
of k discrete plates,
at their
k-
1 interfaces.
cylinders
The
and also have k - 1 discrete sources at the interfaces.
sources,
ally have an arbitrary initial mutual
external
through
two different
boundaries
of a aubatitution
with
arbitrary
that reduces
with homogeneous
external
temperature
distribution,
two different,
constant $lm the problem
boundary
of a Vodicka type of orthogonality
or spheres,
composite
media
each of different have distributed
The composite media addition-
and are exchanging heat at their
arbitrary
time-dependent
surroundings
by means a partial differential equation The solution ia$nally developed by means
coeflcienta.
The solution
is obtained
to that of solving
conditions. relationship.
Notation
c CPi
D, D,
30) G,(r)
h,,: i,j,k Ki
-h(r) 1nl m,n N,
q,(t)
determinant,s thermal diffusivity in the ith section, fV/hr constants constants elements of a determinant determinants constant specific heat at constant pressure in the ith section, Btu/lb, determinant function of time derived function of r constant surface film coefficients, Btu/hr ft2 “R integers thermal conductivity in the ith section, Btu/hr ft OR function of r constant integers “norm”, sum of weighted integrals constant distributed source in the ith section, Btu/hr fts discrete source at the ith junction, Btu/hr ft2 function of time spatial coordinate, ft temperature of surroundings, “R time, hr
2:,T:, t
453
“R
M. H. Cobble derived temperature in the ith section, “R function of time initial condition in the ith section, OR constant eigenfunction, dimensionless temperature in the ith section, “R term in matrix equation Laplacifm operator Kronecker delta integers integer function ratio of thermal conductivities, dimensionless eigenvalue, l/se& density in the ith section, lb,/ft3 derived eigenfunction terms, dimensionless
Introduction
Although problems in conduction heat transfer in composite regions have been treated using complex variable methods and residue theory, such as in Carslaw and Jaeger (l), the first real breakthrough in solving problems of this type occurred in two classic papers of Vodicka (2, 3) in 1950 and 1955. Tittle (4) independently developed the same method in 1965. These solutions are based on a new type of orthogonality relationship. This method for solving heat conduction problems with distributed heat generation in a composite domain of k sections for plates, cylinders or spheres that are insulated on one external boundary, and exchange heat with a single time-dependent surrounding on the other external boundary, is discussed in (5). Beach (6) treated conduction heat transfer in multi-layered cylinders having simple external boundary conditions. In this paper, a much more general problem for plates, cylinders and spheres is solved. The composite media have an arbitrary initial temperature distribution, distributed sources and k - 1 discrete arbitrary, time-dependent sources located at the interfaces. The composite media are exchanging heat at their mutual external boundaries with two different arbitrary, timedependent surroundings, each having a different arbitrary, constant film coefficient through which heat is transferred. Internally, across the k - 1 interfaces, a temperature equality in each region at the interface is maintained, and the difference in heat flux in each region at the interface is equal to the discrete source strength. The solution is developed by substituting a new dependent variable which leads to a new linear partial differential equation having homogeneous boundary conditions. The solution for this derived partial differential equation is obtained using the Vodicka type of orthogonality relationship. When the discrete sources and the surroundings go to zero, the problem reduces to a Vodicka (2) type. If the discrete sources are zero and one film coefficient goes to zero, the problem reduces to a Bulavin-Kashcheev (5)
454
Journal of
The Franklin
Institute
Heat Transfer in Composite Media problem. Additionally, the method reduces to a standard Sturm-Liouville problem when k = 1, giving classical problems for a single component. Statement
of the Problem
The equation for the temperature distribution solidly joined plates, cylinders or spheres is
v2qr, t) + g (r, t) = i
ri d r 6
in the ith section
J$z (r, t),
(1)
z
i = 1,2,3 ,..., k
r&l,
of k
t>o,
J
where V2 = Laplacian operator, Yi = temperature in the ith section, “R, Qi = distributed source in the ith section, Btu/hr ft3, Ki = thermal conductivity in the ith section, Btu/hr ft “R, at = -
Ki
= thermal diffusivity
in the ith section, ftP/b.r,
pi = %zity in the ith section, lb,/ft3, cPi = specific heat in the ith section, Btu/lb,“R. The composite media (plates, cylinders or spheres) are subject to the initial condition : q(r,O) = K(r), ri
media are subject to the external boundary 4
Kk
2
3
(rr, t) = h,[Y,(r,, t) -
conditions:
T,Wl,
(3)
(rk+ljt) = - ~2F’i(rk+l,t) - T,Wl,
(4)
where h,, h, = surface film coefficients, 0 < h,,h,,
Tl, T2 = temperature The composite
of surroundings,
“R.
media are subject to the internal boundary yi(ri+r, t) = Yi+l(rr+l,t),
& 2
Btu/hr ft2 “R,
(ri+17 t) -
&+I
aY. 2
conditions:
i = 1,2,3, . . . . k- 1, i=
(ri+I, t) = &i(t),
(5)
1,2,3 ,..., k-l,
(6)
where Q, = discrete source at the ith junction, TO obtain homogeneous ?(r,t)
Vol.
=
boundary
k+l Vi(r,t)+~~~Lij(r)Elj(t),
ZQO,No.5,November1970
conditions, ri
Btu/hr ft2.
let i = 1,2,3,...,k,
t>O,
(7)
455
M. H. Cobble and, further, let VU&)
= 0
(8)
so that &j(Y) = Aij e(r) + Bij, i=
ri
1,2,3 ,..., k,
j=l,2,3
O(r) is given in Table I for the three geometries.
,..., k+l.
Substitution
(9)
of Eqs. (7) and
I
TABLE Geometry
&“)
Plats Cylinder Sphere
L l/r
(8) into Eq. (1) gives c&$ v2 UJr, t) + g
i
(?-,t)] = 2
(r, t) + F,j@)
F;(t), ,..., k,
l’i
ta0,
(10)
and where F,(t) = -3
T,(t),
(11)
1 2$(t)
Ek+1w= Initial condition,
j = 2,3,4,
f&(t),
=
.. ..
k,
(12)
2k T,(t).
(13)
Ui(r, t) : kfl
V,(r,O)=~(r)-j~l&j(r)F,(0), External boundary
conditions, 2
T~<~
1,2,3 ,..., k.
(14)
UJr, t) : (Yi, t) -3
Lpi,
t) = 0,
(15)
1
2
(Tli+l,
t) +g
Uk(TkI1,
t)
=
(16)
0.
k
Internal boundary
conditions,
q(ri+l,t)
U$r, t) :
= q+l(ri+l,t),
i = 1,2,3 ,..., k-l,
a&+, (r,+,,t), ar
456
(17)
i = 1,2,3 ,..., k-l.
Journal
of The Franklin
(18)
Institute
Heat Transfer in Composite Media External
boundary
Lij(r) :
conditions,
Llj(r,)-~
L&J
= Slj,
j = 1,2,3 ,..., k+l,
(19)
1
L&,+,)
LLj(rk+l ) +g
Internal boundary
&
i = 1,2,3, . . ., k-
L;j(ri+l)-Ki+lL;+l,~(ri+l)
(20)
Lij(r) :
conditions,
Lij(r,+l) = Li+l,j(ri+l),
= 8rc+l,i, j = 1,2,3, . . .) k + 1.
=
1,
j = 1,2,3, . . . . kf
1,
(21)
6i+l,j9
1,2,3 ,..., k-l,
i=
j=
1,2,3 ,..., k+l,
(22)
and where 1,
i =j,
i 0,
i #j.
aij =
(23)
Solution, Lil(r) When we use Eq. (9) in the boundary conditions (external and internal) we get a system of 2k linear non-homogeneous equations for determining the 2k constants Ai, and Bi,. Using the matrix form, the 2k simultaneous equations can be written as equation (24) on page 458. Using matrix notation, these equations may be written as
Mb> =
(25)
{P>,
where the elements af5, f, 5 = 1,2,3, . . . . 2k, are obvious and 21 = Ali,
22 =
Bll,
23 =
A,,,
24 =
B21,
25 =
A,,,
23 =
B,,,
... z 27-l -
\
(26)
...
Avl,
...
zzrl= Bvl,
7 = 1,2,3, . . . . k,
...
z 2k-_1= Am
%x: = -&I,
!
and also P, = 1, p, = 0,
7 = 2,3,4 ,..., 2k.
Now, referring to Eq. (25), if the determinant
lal =
a11
al2
......
a21
a22
......
. a2kl
Vol.290,No.5, November1970
a21c,2 ......
(27)
I
of this system a1,2k a2,2k
. a2k,2k
(#O, /
(28)
457
M. H. Cobble
0
000000
0
000000
+
I
‘= f
c &
0
000000
:
0
000000
i g
‘;
:.
:.
.:
.:
.:
0
0000~0
0
oooo~-;
.:
: .
0
0
r(
i’
r<”
: .
0
: .
:
0
0
0
:
0
0
0
:
0
0
0
0
;
0
0
0
0
:
0
0
0
0
:
0
0
0
:
0
0
0
:
0
0
0
-w
0
458
H
O0
I
0
-
0
Journal of The Franklin
Institute
Heat Transfer in Composite Media then the system of equations has a unique solution given by z1
=!A
[al’
z2
=
la,l ]a(’
z+, .... r,,J#,
***’
(29)
where \a, 1is the determinant formed by replacing the elements (al, azrlas9 . . . azk,J of the 7th column by the column (pr pz p, . . . p,,), respectively. In this way all the Ai, and Ba are found. j = 2,3,4, . . ., k
Solution, L,(r),
The development is the same as Lil(r). identical manner, except that in this case p, = 0,
The argument
7 = 1,2,3 ,..., 2j-2,2j j = 2,3,4,
pzi_1 = l/K. O’(r,),
The result is that, as above, all the A,
proceeds
in an
,..., 2k,
. . . . k.
(30)
I
and Bii, j = 2,3,4, . . ., k, are found.
Solution, Lf,k+l(r) The development is also the same as Lil(r). The argument proceeds in a like manner, except that in this case pn=O, P2k =
~=1,2,3
,..., 2k-1, (31)
I
1.
The result is that all the Ai,k+l and Bi,k+l are found. Solution, Ui(r, t) The solution to Eq. (10) is found by assuming the solution in the form of a series Ui(r, t) = ~~lUin(t)&(r)Y Using an internal boundary ui(ri+r, t) = 5 %(t) n=1
Equation
condition,
X,,(ri+r)
(33) is still an identity,
,..., k,
t>O.
(32)
such as Eq. (17) and Eq. (32), we get
= &+Ari+r, t) = 5 Ui+r,&) X,+l,,(ri+l). n=1
(33)
if for any n
Ui&) Xi&+J For arbitrary
i=l,2,3
ri < r 6 rifl,
= ui+&)
X,+,,(ri+J.
(34)
t and fixed ri+l, Eq. (34) can only be an identity %,V) = %+1,?%(t)= %$).
if (35)
Thus, Eq. (32) may be replaced by &(r, t) = nYIGt) L(r),
ri 6 r < rd+l,
where u,(t) is a function to be determined
Vol.290,No.5,November1970
i=l,2,3
,..., k,
t>O,
(36)
from using the initial conditions,
459
M. H. Cobble and the functions Xi%(r),ri < r 6 T~+~,i = 1,2,3, . . . , k, following eigenvalue problem :
eigenfunctions
of the
where c is 0, 1 or 2 depending on the coordinate system. Equation (37) has the following external boundary conditions :
x;,(rl) -
2 X1n(rJ= 0,
(33)
1
(39)
G&“k+l) + f Xkn(~k+l)= 0, k
and the following internal boundary Xiki+J
= Xi+&-i+J,
conditions : i = 1,273, . . . . k- I,
(40)
= K,+i X;+l,n(ri+l), i = 1,2,3, . . . , k - 1,
& X;,(ri+,)
(41)
and where pm is an eigenvalue of the problem. The solution to Eq. (37) is n = 0,1,2, a**,
&L(r) = ~dnPin(~)+&VMr),
(42)
where the specific terms are shown in Table II. TABLE II Geometry
C
PM3
0
n
#incr)
P7im
0
T
1
1,2,3,
0 Cylinder
1
0
lnr
1
1,2,3,
1 Sphere
...
0
l/r
2
...
1,2,3, . . .
2
If we substitute Eq. (42) in the boundary conditions (38)-(41), we get a system of 2k linear homogeneous equations for determining the 2k integration constants Ai, and Bin. In the theory of linear homogeneous equations, this system of equations has a non-trivial solution when its determinant is equal to zero. Setting this determinant equal to zero gives the transcendental equation
&(r,) D=D,=
.. .
460
...
. ..
.
,
. ..
. ..
. .. =O,
n=1,2,3
,...,
(43)
~,k(rk+l)
Journal
of The Franklin
Institute
Heat Transfer in Composite Media
and D, # 0,
n = 0
(p. = 0 is not an eigenvalue)
(44)
for determining the eigenvalues pn. An expanded form of this determinant for any I%is shown in the Appendix. Equation (43) with the accompanying Eq. (42) has an infinite number of roots, #LL1
. ..)
(45)
and for each of these roots, there are corresponding values of Ai, and Bi, in the eigenfunctions Xin(r), i = 1,2,3, . . . . k, which can be determined to within a multiple of an arbitrary constant, and due to the form of Eq. (36) and a following normalization process, the solution is unaffected by the value of this constant, and with complete generality, it can be replaced by the number 1 (one), and Xin(r), i = 1,2,3, . . . . k, is then completely specified.* The functions Xim(r) and Xin(r), ri 6 r < T{+~,i = 1,2,3, . . . , k, m, n integers, are orthogonal in the domain [rr, rk+J. The orthogonality condition of these functions is
5 picpi
i=l
“+lrcX&r) Xin(r)dr =
(46)
s ri
This identity can be shown by integration by parts, and then using the internal and external boundary conditions. Since the function L,(r) satisfies Dirichlet’s conditions, and if Q&T,t) and G&r) do, we expand these functions in an in&rite series of the eigenfunctions
&iP, 4 Pi $2
a, = nYlcm Xi?&%
(49)
Multiplying the left- and right-hand sides of (47)-( 49) by pi cpi rc Xim(r) and integrating with respect to r from ri to ri+l and summing each of the identities for all the values of i, and using the orthogonality relation (46), we obtain
9, =
1 N,
k
s
%zlPi %i ri“+Irc GJr) Xi%(r)dr,
1 k Tii+l Q&) =~5;1~-.r II rc&&,t)X,,(r)dr, s
n=
1,2,3 ,...,
n = 1,2,3,...,
(59)
(52)
* The presentation closely parallels that of Bulavin and Kashcheev (7).
Vol.290,No.5,November1970
461
M. H.
Cobble
and where n = 1,2,3, . . . .
N, = ~~~ic~cS:“ZXg(r)dr, Substituting
Eqs. (36), (48) and (49), and using (37) in Eq. (lo), we obtain
f.“i <
Equation
(53)
r G rg+1,
i=l,2,3
,..., k,
t>O.
(54)
n = 1,2,3, . . . .
(55)
(54) is satisfied if %
= -~,&(t)+p,(t),
(t)+&%&)
Using Eqs. (36), (47) and (14), we get the initial condition k+l
u,(O)=g,=-CZ,j~~(0)+v,, i=l
n=l,2,3
,...,
(56)
where 1 k va = N, 4z1P$ $i
s
+i‘“‘P”l$(r)
Xin(r) dr,
The solution to Eq. (55) using the initial condition
n = 1,2,3, . . . .
(57)
is
kfl %V)
=
g,exP
(-
~$4
-
jzlLjP;(t)*
exp
t-p”,
t) + q,(t)*
e=p
( --pit),
n = 1,2,3 ,...,
(58)
where the symbol * denotes convolution. Thus, the solution for Ud(r,t) is
q(r, t) = g
n=l
gn exp ( -pi
t) - yZnj St1
‘F;(T) exp [ - &(t - T)] d7 s 0
(59)
In Eq. (59), Xin(T) = Ai~PS,(r) +Bin$4,(r),
n = l, 2, 3, ..*)
(60)
where pin and #im are given in Table II. In Eqs. (59) and (60), the following hold
A,, = 1
462
(61)
Journal of The Franklin Institute
Heat Transfer in Composite Media and
B1n=
B2n
hi-s I.,
i =
Ain = Ibl B
=1b2i-11
i=
in
=
(b,l Ibl
A,,
=
’
. . . . k,
1,2,3 ,..., k,
-vi-'
we obtain Ibl and lb,/, 5 = 1,2,3, . . . . 2k- 1 from the elements in the determinant D, the transcendental Eq. (43), which is shown in the general expanded form of order 2k in the Appendix. Using the elements, we may write
D=
b11
b,,
...
...
bI,2x:
b21
b,,
...
...
b2,,,
.
.
b 2k,l
b2rc,2 .a.
..a
= 0
(63)
b2rc.2k
and / b I is given by
lb,
‘f”
=
bf”
‘1’
1b2k-l,2
&--1,s .--
‘27
‘;’
.
. a.9
(64)
#O
hk--1,s
and is of order 2k - 1. The elements are from D. The determinant I b, I, [= 1,2,3 ,..., 2k- 1, is formed by replacing the 5th column in 1b 1, by the column ( -b,, -b,, - b,, . . . - b2k_-l,l)formed from the corresponding elements in D. Solution, &(r, t) All the components of Ui(r, t) and &(r) temperature in each section is given by k+l q(r,t)
=
L$(r,t)+
CL,(r)l$(t), j=l
r,
are now known,
i = 1,2,3,...,k,
and so the
t>O.
(7)
The solution Yi(r, t) gives the temperature distribution in any of the k plates, cylinders or spheres that are solidly joined at the k- 1 interfaces. The composite media (plates, cylinders or spheres) have distributed sources, and k - 1 discrete, time-dependent sources located at the interfaces. Additionally, the composite media have an arbitrary initial temperature distribution and are also subject to exchanging heat at their mutual external boundaries with two arbitrary, time-dependent surroundings through two arbitrary, constantfilm coefficients.
Vol.290,No. 5,November 1970
463
M. H. Cobble Appendix The expanded form of the determinant, Eq. (43), that gives the transcendental equation for determining the eigenvalues pn for a solid consisting of k plates, cylinders or spheres is given in Eq. (A.6). In Eq. (A.6) the following
(A.11 (A4 (A-3)
(A.4)
(A.5) 0 0 0 0 0
0 0 0
0
0
0
0
0
0
0
...
...
...
...
. . .
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0
...
0
0
...
0
0
...
0
0
...
0
-&r,)::: -&b4) ... 0 0 0
464
*-* . .. . .. .. . ...
Mr4)
x
-P&,) --~7;k& ... 0 0 0
(r 4
3&
0
0. i.6)
0 0 ...
#k-1,&-k)
Pk-l,?@k) h--l
f&n(%)
0
#k-l
Y)&drk)
0
-
p3knb.k)
-
$kn(r,)
-
S?hb”k)
-
&&k)
‘&&“k+l)
Journal
@drk+l)
of The Franklin
Institute
Heat Transfer in Composite Media References
(1) H. S. Carslaw and J. C. Jaeger, “Conduction of Heat in Solids”, 2nd ed., pp. 305-307, (2) V.
Oxford, Clarendon Press, 1959. Vodicka, “Warmeleitung in
geschichteten
Kugel
und
Zylinderkiirpern”,
Id. Chem. Engng, Vol. 5, No. 1, pp. 112-115, 1965. (6) H. L. Beach, “The application of the orthogonal expansion technique to conduction heat transfer problems in multileyer cylinders”, M.S. Thesis, Mech. and Aerospace Eng., N.C. State University, Raleigh, N.C., 1967. (7) P. E. Bulavin and V. M. Keshcheev, “Solution of the non-homogeneous heat conduction equation for multilayered bodies”, Int. Chem. Engng, Vol. 5, No. 1, pp. 112-115, 1965.
Vol. 290, No. 5. November
1970
465
Transfer
Distributed
in Composite Media Subject to
Sources, and Time-Dependent
Discrete
Sources and Surroundings br
M. H.
COBBLE
Department of Mechanical Engineering New Mexico State Universip, Las Cruces, New Mexico The equation for the temperature
ABSTRACT:
is solved.
The composite
material
solidly
joined
distribution in each of k sections of a composite
consists
of k discrete plates,
at their
k-
1 interfaces.
cylinders
The
and also have k - 1 discrete sources at the interfaces.
sources,
ally have an arbitrary initial mutual
external
through
two different
boundaries
of a aubatitution
with
arbitrary
that reduces
with homogeneous
external
temperature
distribution,
two different,
constant $lm the problem
boundary
of a Vodicka type of orthogonality
or spheres,
composite
media
each of different have distributed
The composite media addition-
and are exchanging heat at their
arbitrary
time-dependent
surroundings
by means a partial differential equation The solution ia$nally developed by means
coeflcienta.
The solution
is obtained
to that of solving
conditions. relationship.
Notation
c CPi
D, D,
30) G,(r)
h,,: i,j,k Ki
-h(r) 1nl m,n N,
q,(t)
determinant,s thermal diffusivity in the ith section, fV/hr constants constants elements of a determinant determinants constant specific heat at constant pressure in the ith section, Btu/lb, determinant function of time derived function of r constant surface film coefficients, Btu/hr ft2 “R integers thermal conductivity in the ith section, Btu/hr ft OR function of r constant integers “norm”, sum of weighted integrals constant distributed source in the ith section, Btu/hr fts discrete source at the ith junction, Btu/hr ft2 function of time spatial coordinate, ft temperature of surroundings, “R time, hr
2:,T:, t
453
“R
M. H. Cobble derived temperature in the ith section, “R function of time initial condition in the ith section, OR constant eigenfunction, dimensionless temperature in the ith section, “R term in matrix equation Laplacifm operator Kronecker delta integers integer function ratio of thermal conductivities, dimensionless eigenvalue, l/se& density in the ith section, lb,/ft3 derived eigenfunction terms, dimensionless
Introduction
Although problems in conduction heat transfer in composite regions have been treated using complex variable methods and residue theory, such as in Carslaw and Jaeger (l), the first real breakthrough in solving problems of this type occurred in two classic papers of Vodicka (2, 3) in 1950 and 1955. Tittle (4) independently developed the same method in 1965. These solutions are based on a new type of orthogonality relationship. This method for solving heat conduction problems with distributed heat generation in a composite domain of k sections for plates, cylinders or spheres that are insulated on one external boundary, and exchange heat with a single time-dependent surrounding on the other external boundary, is discussed in (5). Beach (6) treated conduction heat transfer in multi-layered cylinders having simple external boundary conditions. In this paper, a much more general problem for plates, cylinders and spheres is solved. The composite media have an arbitrary initial temperature distribution, distributed sources and k - 1 discrete arbitrary, time-dependent sources located at the interfaces. The composite media are exchanging heat at their mutual external boundaries with two different arbitrary, timedependent surroundings, each having a different arbitrary, constant film coefficient through which heat is transferred. Internally, across the k - 1 interfaces, a temperature equality in each region at the interface is maintained, and the difference in heat flux in each region at the interface is equal to the discrete source strength. The solution is developed by substituting a new dependent variable which leads to a new linear partial differential equation having homogeneous boundary conditions. The solution for this derived partial differential equation is obtained using the Vodicka type of orthogonality relationship. When the discrete sources and the surroundings go to zero, the problem reduces to a Vodicka (2) type. If the discrete sources are zero and one film coefficient goes to zero, the problem reduces to a Bulavin-Kashcheev (5)
454
Journal of
The Franklin
Institute
Heat Transfer in Composite Media problem. Additionally, the method reduces to a standard Sturm-Liouville problem when k = 1, giving classical problems for a single component. Statement
of the Problem
The equation for the temperature distribution solidly joined plates, cylinders or spheres is
v2qr, t) + g (r, t) = i
ri d r 6
in the ith section
J$z (r, t),
(1)
z
i = 1,2,3 ,..., k
r&l,
of k
t>o,
J
where V2 = Laplacian operator, Yi = temperature in the ith section, “R, Qi = distributed source in the ith section, Btu/hr ft3, Ki = thermal conductivity in the ith section, Btu/hr ft “R, at = -
Ki
= thermal diffusivity
in the ith section, ftP/b.r,
pi = %zity in the ith section, lb,/ft3, cPi = specific heat in the ith section, Btu/lb,“R. The composite media (plates, cylinders or spheres) are subject to the initial condition : q(r,O) = K(r), ri
media are subject to the external boundary 4
Kk
2
3
(rr, t) = h,[Y,(r,, t) -
conditions:
T,Wl,
(3)
(rk+ljt) = - ~2F’i(rk+l,t) - T,Wl,
(4)
where h,, h, = surface film coefficients, 0 < h,,h,,
Tl, T2 = temperature The composite
of surroundings,
“R.
media are subject to the internal boundary yi(ri+r, t) = Yi+l(rr+l,t),
& 2
Btu/hr ft2 “R,
(ri+17 t) -
&+I
aY. 2
conditions:
i = 1,2,3, . . . . k- 1, i=
(ri+I, t) = &i(t),
(5)
1,2,3 ,..., k-l,
(6)
where Q, = discrete source at the ith junction, TO obtain homogeneous ?(r,t)
Vol.
=
boundary
k+l Vi(r,t)+~~~Lij(r)Elj(t),
ZQO,No.5,November1970
conditions, ri
Btu/hr ft2.
let i = 1,2,3,...,k,
t>O,
(7)
455
M. H. Cobble and, further, let VU&)
= 0
(8)
so that &j(Y) = Aij e(r) + Bij, i=
ri
1,2,3 ,..., k,
j=l,2,3
O(r) is given in Table I for the three geometries.
,..., k+l.
Substitution
(9)
of Eqs. (7) and
I
TABLE Geometry
&“)
Plats Cylinder Sphere
L l/r
(8) into Eq. (1) gives c&$ v2 UJr, t) + g
i
(?-,t)] = 2
(r, t) + F,j@)
F;(t), ,..., k,
l’i
ta0,
(10)
and where F,(t) = -3
T,(t),
(11)
1 2$(t)
Ek+1w= Initial condition,
j = 2,3,4,
f&(t),
=
.. ..
k,
(12)
2k T,(t).
(13)
Ui(r, t) : kfl
V,(r,O)=~(r)-j~l&j(r)F,(0), External boundary
conditions, 2
T~<~
1,2,3 ,..., k.
(14)
UJr, t) : (Yi, t) -3
Lpi,
t) = 0,
(15)
1
2
(Tli+l,
t) +g
Uk(TkI1,
t)
=
(16)
0.
k
Internal boundary
conditions,
q(ri+l,t)
U$r, t) :
= q+l(ri+l,t),
i = 1,2,3 ,..., k-l,
a&+, (r,+,,t), ar
456
(17)
i = 1,2,3 ,..., k-l.
Journal
of The Franklin
(18)
Institute
Heat Transfer in Composite Media External
boundary
Lij(r) :
conditions,
Llj(r,)-~
L&J
= Slj,
j = 1,2,3 ,..., k+l,
(19)
1
L&,+,)
LLj(rk+l ) +g
Internal boundary
&
i = 1,2,3, . . ., k-
L;j(ri+l)-Ki+lL;+l,~(ri+l)
(20)
Lij(r) :
conditions,
Lij(r,+l) = Li+l,j(ri+l),
= 8rc+l,i, j = 1,2,3, . . .) k + 1.
=
1,
j = 1,2,3, . . . . kf
1,
(21)
6i+l,j9
1,2,3 ,..., k-l,
i=
j=
1,2,3 ,..., k+l,
(22)
and where 1,
i =j,
i 0,
i #j.
aij =
(23)
Solution, Lil(r) When we use Eq. (9) in the boundary conditions (external and internal) we get a system of 2k linear non-homogeneous equations for determining the 2k constants Ai, and Bi,. Using the matrix form, the 2k simultaneous equations can be written as equation (24) on page 458. Using matrix notation, these equations may be written as
Mb> =
(25)
{P>,
where the elements af5, f, 5 = 1,2,3, . . . . 2k, are obvious and 21 = Ali,
22 =
Bll,
23 =
A,,,
24 =
B21,
25 =
A,,,
23 =
B,,,
... z 27-l -
\
(26)
...
Avl,
...
zzrl= Bvl,
7 = 1,2,3, . . . . k,
...
z 2k-_1= Am
%x: = -&I,
!
and also P, = 1, p, = 0,
7 = 2,3,4 ,..., 2k.
Now, referring to Eq. (25), if the determinant
lal =
a11
al2
......
a21
a22
......
. a2kl
Vol.290,No.5, November1970
a21c,2 ......
(27)
I
of this system a1,2k a2,2k
. a2k,2k
(#O, /
(28)
457
M. H. Cobble
0
000000
0
000000
+
I
‘= f
c &
0
000000
:
0
000000
i g
‘;
:.
:.
.:
.:
.:
0
0000~0
0
oooo~-;
.:
: .
0
0
r(
i’
r<”
: .
0
: .
:
0
0
0
:
0
0
0
:
0
0
0
0
;
0
0
0
0
:
0
0
0
0
:
0
0
0
:
0
0
0
:
0
0
0
-w
0
458
H
O0
I
0
-
0
Journal of The Franklin
Institute
Heat Transfer in Composite Media then the system of equations has a unique solution given by z1
=!A
[al’
z2
=
la,l ]a(’
z+, .... r,,J#,
***’
(29)
where \a, 1is the determinant formed by replacing the elements (al, azrlas9 . . . azk,J of the 7th column by the column (pr pz p, . . . p,,), respectively. In this way all the Ai, and Ba are found. j = 2,3,4, . . ., k
Solution, L,(r),
The development is the same as Lil(r). identical manner, except that in this case p, = 0,
The argument
7 = 1,2,3 ,..., 2j-2,2j j = 2,3,4,
pzi_1 = l/K. O’(r,),
The result is that, as above, all the A,
proceeds
in an
,..., 2k,
. . . . k.
(30)
I
and Bii, j = 2,3,4, . . ., k, are found.
Solution, Lf,k+l(r) The development is also the same as Lil(r). The argument proceeds in a like manner, except that in this case pn=O, P2k =
~=1,2,3
,..., 2k-1, (31)
I
1.
The result is that all the Ai,k+l and Bi,k+l are found. Solution, Ui(r, t) The solution to Eq. (10) is found by assuming the solution in the form of a series Ui(r, t) = ~~lUin(t)&(r)Y Using an internal boundary ui(ri+r, t) = 5 %(t) n=1
Equation
condition,
X,,(ri+r)
(33) is still an identity,
,..., k,
t>O.
(32)
such as Eq. (17) and Eq. (32), we get
= &+Ari+r, t) = 5 Ui+r,&) X,+l,,(ri+l). n=1
(33)
if for any n
Ui&) Xi&+J For arbitrary
i=l,2,3
ri < r 6 rifl,
= ui+&)
X,+,,(ri+J.
(34)
t and fixed ri+l, Eq. (34) can only be an identity %,V) = %+1,?%(t)= %$).
if (35)
Thus, Eq. (32) may be replaced by &(r, t) = nYIGt) L(r),
ri 6 r < rd+l,
where u,(t) is a function to be determined
Vol.290,No.5,November1970
i=l,2,3
,..., k,
t>O,
(36)
from using the initial conditions,
459
M. H. Cobble and the functions Xi%(r),ri < r 6 T~+~,i = 1,2,3, . . . , k, following eigenvalue problem :
eigenfunctions
of the
where c is 0, 1 or 2 depending on the coordinate system. Equation (37) has the following external boundary conditions :
x;,(rl) -
2 X1n(rJ= 0,
(33)
1
(39)
G&“k+l) + f Xkn(~k+l)= 0, k
and the following internal boundary Xiki+J
= Xi+&-i+J,
conditions : i = 1,273, . . . . k- I,
(40)
= K,+i X;+l,n(ri+l), i = 1,2,3, . . . , k - 1,
& X;,(ri+,)
(41)
and where pm is an eigenvalue of the problem. The solution to Eq. (37) is n = 0,1,2, a**,
&L(r) = ~dnPin(~)+&VMr),
(42)
where the specific terms are shown in Table II. TABLE II Geometry
C
PM3
0
n
#incr)
P7im
0
T
1
1,2,3,
0 Cylinder
1
0
lnr
1
1,2,3,
1 Sphere
...
0
l/r
2
...
1,2,3, . . .
2
If we substitute Eq. (42) in the boundary conditions (38)-(41), we get a system of 2k linear homogeneous equations for determining the 2k integration constants Ai, and Bin. In the theory of linear homogeneous equations, this system of equations has a non-trivial solution when its determinant is equal to zero. Setting this determinant equal to zero gives the transcendental equation
&(r,) D=D,=
.. .
460
...
. ..
.
,
. ..
. ..
. .. =O,
n=1,2,3
,...,
(43)
~,k(rk+l)
Journal
of The Franklin
Institute
Heat Transfer in Composite Media
and D, # 0,
n = 0
(p. = 0 is not an eigenvalue)
(44)
for determining the eigenvalues pn. An expanded form of this determinant for any I%is shown in the Appendix. Equation (43) with the accompanying Eq. (42) has an infinite number of roots, #LL1
. ..)
(45)
and for each of these roots, there are corresponding values of Ai, and Bi, in the eigenfunctions Xin(r), i = 1,2,3, . . . . k, which can be determined to within a multiple of an arbitrary constant, and due to the form of Eq. (36) and a following normalization process, the solution is unaffected by the value of this constant, and with complete generality, it can be replaced by the number 1 (one), and Xin(r), i = 1,2,3, . . . . k, is then completely specified.* The functions Xim(r) and Xin(r), ri 6 r < T{+~,i = 1,2,3, . . . , k, m, n integers, are orthogonal in the domain [rr, rk+J. The orthogonality condition of these functions is
5 picpi
i=l
“+lrcX&r) Xin(r)dr =
(46)
s ri
This identity can be shown by integration by parts, and then using the internal and external boundary conditions. Since the function L,(r) satisfies Dirichlet’s conditions, and if Q&T,t) and G&r) do, we expand these functions in an in&rite series of the eigenfunctions
&iP, 4 Pi $2
a, = nYlcm Xi?&%
(49)
Multiplying the left- and right-hand sides of (47)-( 49) by pi cpi rc Xim(r) and integrating with respect to r from ri to ri+l and summing each of the identities for all the values of i, and using the orthogonality relation (46), we obtain
9, =
1 N,
k
s
%zlPi %i ri“+Irc GJr) Xi%(r)dr,
1 k Tii+l Q&) =~5;1~-.r II rc&&,t)X,,(r)dr, s
n=
1,2,3 ,...,
n = 1,2,3,...,
(59)
(52)
* The presentation closely parallels that of Bulavin and Kashcheev (7).
Vol.290,No.5,November1970
461
M. H.
Cobble
and where n = 1,2,3, . . . .
N, = ~~~ic~cS:“ZXg(r)dr, Substituting
Eqs. (36), (48) and (49), and using (37) in Eq. (lo), we obtain
f.“i <
Equation
(53)
r G rg+1,
i=l,2,3
,..., k,
t>O.
(54)
n = 1,2,3, . . . .
(55)
(54) is satisfied if %
= -~,&(t)+p,(t),
(t)+&%&)
Using Eqs. (36), (47) and (14), we get the initial condition k+l
u,(O)=g,=-CZ,j~~(0)+v,, i=l
n=l,2,3
,...,
(56)
where 1 k va = N, 4z1P$ $i
s
+i‘“‘P”l$(r)
Xin(r) dr,
The solution to Eq. (55) using the initial condition
n = 1,2,3, . . . .
(57)
is
kfl %V)
=
g,exP
(-
~$4
-
jzlLjP;(t)*
exp
t-p”,
t) + q,(t)*
e=p
( --pit),
n = 1,2,3 ,...,
(58)
where the symbol * denotes convolution. Thus, the solution for Ud(r,t) is
q(r, t) = g
n=l
gn exp ( -pi
t) - yZnj St1
‘F;(T) exp [ - &(t - T)] d7 s 0
(59)
In Eq. (59), Xin(T) = Ai~PS,(r) +Bin$4,(r),
n = l, 2, 3, ..*)
(60)
where pin and #im are given in Table II. In Eqs. (59) and (60), the following hold
A,, = 1
462
(61)
Journal of The Franklin Institute
Heat Transfer in Composite Media and
B1n=
B2n
hi-s I.,
i =
Ain = Ibl B
=1b2i-11
i=
in
=
(b,l Ibl
A,,
=
’
. . . . k,
1,2,3 ,..., k,
-vi-'
we obtain Ibl and lb,/, 5 = 1,2,3, . . . . 2k- 1 from the elements in the determinant D, the transcendental Eq. (43), which is shown in the general expanded form of order 2k in the Appendix. Using the elements, we may write
D=
b11
b,,
...
...
bI,2x:
b21
b,,
...
...
b2,,,
.
.
b 2k,l
b2rc,2 .a.
..a
= 0
(63)
b2rc.2k
and / b I is given by
lb,
‘f”
=
bf”
‘1’
1b2k-l,2
&--1,s .--
‘27
‘;’
.
. a.9
(64)
#O
hk--1,s
and is of order 2k - 1. The elements are from D. The determinant I b, I, [= 1,2,3 ,..., 2k- 1, is formed by replacing the 5th column in 1b 1, by the column ( -b,, -b,, - b,, . . . - b2k_-l,l)formed from the corresponding elements in D. Solution, &(r, t) All the components of Ui(r, t) and &(r) temperature in each section is given by k+l q(r,t)
=
L$(r,t)+
CL,(r)l$(t), j=l
r,
are now known,
i = 1,2,3,...,k,
and so the
t>O.
(7)
The solution Yi(r, t) gives the temperature distribution in any of the k plates, cylinders or spheres that are solidly joined at the k- 1 interfaces. The composite media (plates, cylinders or spheres) have distributed sources, and k - 1 discrete, time-dependent sources located at the interfaces. Additionally, the composite media have an arbitrary initial temperature distribution and are also subject to exchanging heat at their mutual external boundaries with two arbitrary, time-dependent surroundings through two arbitrary, constantfilm coefficients.
Vol.290,No. 5,November 1970
463
M. H. Cobble Appendix The expanded form of the determinant, Eq. (43), that gives the transcendental equation for determining the eigenvalues pn for a solid consisting of k plates, cylinders or spheres is given in Eq. (A.6). In Eq. (A.6) the following
(A.11 (A4 (A-3)
(A.4)
(A.5) 0 0 0 0 0
0 0 0
0
0
0
0
0
0
0
...
...
...
...
. . .
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0
...
0
0
...
0
0
...
0
0
...
0
-&r,)::: -&b4) ... 0 0 0
464
*-* . .. . .. .. . ...
Mr4)
x
-P&,) --~7;k& ... 0 0 0
(r 4
3&
0
0. i.6)
0 0 ...
#k-1,&-k)
Pk-l,?@k) h--l
f&n(%)
0
#k-l
Y)&drk)
0
-
p3knb.k)
-
$kn(r,)
-
S?hb”k)
-
&&k)
‘&&“k+l)
Journal
@drk+l)
of The Franklin
Institute
Heat Transfer in Composite Media References
(1) H. S. Carslaw and J. C. Jaeger, “Conduction of Heat in Solids”, 2nd ed., pp. 305-307, (2) V.
Oxford, Clarendon Press, 1959. Vodicka, “Warmeleitung in
geschichteten
Kugel
und
Zylinderkiirpern”,
Id. Chem. Engng, Vol. 5, No. 1, pp. 112-115, 1965. (6) H. L. Beach, “The application of the orthogonal expansion technique to conduction heat transfer problems in multileyer cylinders”, M.S. Thesis, Mech. and Aerospace Eng., N.C. State University, Raleigh, N.C., 1967. (7) P. E. Bulavin and V. M. Keshcheev, “Solution of the non-homogeneous heat conduction equation for multilayered bodies”, Int. Chem. Engng, Vol. 5, No. 1, pp. 112-115, 1965.
Vol. 290, No. 5. November
1970
465